On two multistable extensions of stable Lévy motion and their semimartingale representation

Ronan Le Guével (1) , Jacques Lévy-Vehel (2, 3) , Lining Liu (2)

1 IRMAR - Institut de Recherche Mathématique de Rennes
2 Regularity - Probabilistic modelling of irregularity and application to uncertainties management
3 MAS - Mathématiques Appliquées aux Systèmes - EA 4037

We compare two definitions of multistable Lévy motions. Such processes are extensions of classical Lévy motion where the stability index is allowed to vary in time. We show that the two multistable Lévy motions have distinct properties: in particular, one is a pure-jump Markov process, while the other one satisfies neither of these properties. We prove that both are semimartingales and provide semimartingale decompositions.

Mots clés: Semi-martingale; Multistable process; Lévy motion

Domaines: Probabilités [math.PR]

Ronan Le Guével, Jacques Lévy-Vehel, Lining Liu. On two multistable extensions of stable Lévy motion and their semimartingale representation. Journal of Theoretical Probability, 2015, 28 (3), pp.1125-1144. ⟨10.1007/s10959-013-0528-6⟩. ⟨hal-00730680v2⟩